Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise

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• 作者：C. Bauzet ; J. Charrier ; T. Gallouët
• 关键词：Stochastic PDEs ; First ; order hyperbolic equations ; Young measures ; Monotone finite volume schemes
• 刊名：Stochastic Partial Differential Equations: Analysis and Computations
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：4
• 期：1
• 页码：150-223
• 全文大小：918 KB
• 参考文献：1.Balder, E.J.: Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31(suppl. 1), 1–69 (2000). Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997)MathSciNet MATH
  2.Bauzet, C.: On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments. J. Evolut. Equ. 14(2), 333–356 (2014)MathSciNet CrossRef MATH
  3.Bauzet, C., Charrier, J., Gallouët T.: Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation. Math. Comput. (2014) (submitted)
  4.Bauzet, C., Vallet, G., Wittbold, P.: The Cauchy problem for a conservation law with a multiplicative stochastic perturbation. J. Hyperbolic Diff. Equ. 9(4), 661–709 (2012)MathSciNet CrossRef MATH
  5.Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. J. Funct. Anal. 4(266), 2503–2545 (2014)MathSciNet CrossRef MATH
  6.Biswas, I.H., Majee, A.K.: Stochastic conservation laws: weak-in-time formulation and strong entropy condition. J. Funct. Anal. 7(267), 2199–2252 (2014)MathSciNet CrossRef MATH
  7.Chainais-Hillairet, C.: Second-order finite-volume schemes for a non-linear hyperbolic equation: error estimate. Math. Methods Appl. Sci. 23(5), 467–490 (2000)MathSciNet CrossRef MATH
  8.Chen, G.-Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)MathSciNet CrossRef MATH
  9.Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)CrossRef MATH
  10.Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)MathSciNet CrossRef MATH
  11.Eymard, R., Gallouët, T., Herbin, R.: Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation. Chin. Ann. Math. Ser. B 16(1), 1–14 (1995). A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 1, 119MathSciNet MATH
  12.Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Analysis, vol. VII, Handb. Numer. Anal., VII, pp. 713–1020. North-Holland, Amsterdam (2000)
  13.Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)MathSciNet CrossRef MATH
  14.Grecksch, W., Tudor, C.: Stochastic Evolution Equations. Mathematical Research, vol. 85. Akademie, Berlin (1995). A Hilbert space approachMATH
  15.Hofmanová, M.: Bhatnagar-gross-krook approximation to stochastic scalar conservation laws. Ann. Inst. H. Poincare Probab. Statist. (2014)
  16.Holden, H., Risebro, N.H.: A stochastic approach to conservation laws. In: Third International Conference on Hyperbolic Problems, vol. I, II (Uppsala, 1990), pp. 575–587. Studentlitteratur, Lund (1991)
  17.Kim, J.U.: On the stochastic porous medium equation. J. Diff. Equ. 220(1), 163–194 (2006)MathSciNet CrossRef MATH
  18.Kröker, I., Rohde, C.: Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62(4), 441–456 (2012)MathSciNet CrossRef MATH
  19.Panov, EYu.: On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation. Izv. Ross. Akad. Nauk Ser. Mat. 60(2), 107–148 (1996)MathSciNet CrossRef MATH
  20.Vallet, G.: Stochastic perturbation of nonlinear degenerate parabolic problems. Differential Integral Equations 21(11–12), 1055–1082 (2008)MathSciNet MATH
  
• 作者单位：C. Bauzet (1)
  J. Charrier (1)
  T. Gallouët (1)
  
  1. Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453, Marseille, France
  
• 刊物主题：Probability Theory and Stochastic Processes; Partial Differential Equations; Statistical Theory and Methods; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Numerical Analysis;
• 出版者：Springer US
• ISSN：2194-041X

文摘

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in \(L^{2}(\mathbb {R}^d)\). After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove under a stability condition on the time step the convergence of the finite volume approximation towards the unique stochastic entropy solution of the equation.